Polynomial Form Research Articles

Physical multi-factor driven nonlinear superposition for machining deformation reconstruction

In recent years, the machining deformation of titanium alloy frame parts has become a major research issue in the field of aviation manufacture. Through the generation mechanism, the machining deformation is divided into three parts including bulk residual stresses, cutting forces and other uncertain factors induced deformation. The nonlinearity of physical multi-factor machining deformation is revealed by the comparison between the purely linear addition and coupling superposition. A novel nonlinear superposition model is proposed for machining deformation reconstruction in this work. With the experimental values as simulation inputs, the physical multi-factor driven simulation is developed to solve the nonlinear parameters of machining components in the superposition model. The posterior information embedded in the measurement is utilized to calibrate the nonlinear superposition model in the form of binary polynomials with five orders. The nonlinear superposition model is validated through four series of machining tasks, with the average model accuracy at 91.09%. To quantitively evaluate the proportion share of deformation components, a dimensionless equivalent machining deformation ratio is developed by the integral operation of the established nonlinear model. The results indicate that the bulk residual stresses, cutting forces and other uncertain factors account for percentages of 24.72%, 65.07%, and 13.21% (the ratio among them around 2:5:1). The proposed nonlinear superposition model and dimensionless equivalent ratio in this work display the potential application prospects in the field of machining deformation prediction and control.

AMITE: A Novel Polynomial Expansion for Analyzing Neural Network Nonlinearities.

Polynomial expansions are important in the analysis of neural network nonlinearities. They have been applied thereto addressing well-known difficulties in verification, explainability, and security. Existing approaches span classical Taylor and Chebyshev methods, asymptotics, and many numerical approaches. We find that, while these have useful properties individually, such as exact error formulas, adjustable domain, and robustness to undefined derivatives, there are no approaches that provide a consistent method, yielding an expansion with all these properties. To address this, we develop an analytically modified integral transform expansion (AMITE), a novel expansion via integral transforms modified using derived criteria for convergence. We show the general expansion and then demonstrate an application for two popular activation functions: hyperbolic tangent and rectified linear units. Compared with existing expansions (i.e., Chebyshev, Taylor, and numerical) employed to this end, AMITE is the first to provide six previously mutually exclusive desired expansion properties, such as exact formulas for the coefficients and exact expansion errors. We demonstrate the effectiveness of AMITE in two case studies. First, a multivariate polynomial form is efficiently extracted from a single hidden layer black-box multilayer perceptron (MLP) to facilitate equivalence testing from noisy stimulus-response pairs. Second, a variety of feedforward neural network (FFNN) architectures having between three and seven layers are range bounded using Taylor models improved by the AMITE polynomials and error formulas. AMITE presents a new dimension of expansion methods suitable for the analysis/approximation of nonlinearities in neural networks, opening new directions and opportunities for the theoretical analysis and systematic testing of neural networks.

Information and Theoretical Foundations for Control of the State of Agrocenoses in Real Time

Objective: Presentation of a complete theory of management of the state of agrocenoses, including sowing crops and weeds. This theory makes it possible to obtain a given crop yield with the required reliability. Such management is formed on the basis of estimates of the parameters of the state of crops, soil environment and weeds, formed according to the data of Earth remote sensing in real time. Methods: The presented theory is based on new mathematical models of the parameters of the state of agricultural crops, soil environment, weeds, as well as models of the relationship of these parameters with remote sensing (RS) data. Control factors in agricultural technology are doses of mineral fertilizers, herbicides and irrigation rates. Therefore, the parameters of the state in the management system under consideration, in addition to the parameters of the biomass of agricultural crops and weeds, are the chemical parameters of the soil environment. All technological operations are carried out at the onset of certain phenological phases of the development of crop sowing. RS data are entered precisely at such moments of time when the parameters of the state of crops and weeds are estimated on their basis. The presented theory is based on classical control principles used in modern dynamic systems. The most important feature of the proposed approach to solving the problem of agrocenoses management is the development of the principle of spatial distribution of agricultural technology parameters. It consists in using models with spatial variables, which are complemented by models of spatial approximation of controls in the form of multidimensional power polynomials. The parameters of the approximation models are evaluated according to the optimal parameters of technological operations obtained on 15-20 test sites of the field. Results: In accordance with the proposed theory, mathematical models of agrocenoses, a real-time control algorithm using RS data and a specialized software package were developed, with the help of which the control system was tested on the example of spring wheat sowing. Conclusion: As a result of testing a specialized software package for the implementation of the proposed theory, the operability of the proposed theory of agrocenoses control in real time and the possibility of using it in practical systems for automated control of agricultural technologies were established.

Residual stress prediction across dimensions using improved radial basis function based eigenstrain reconstruction

Accurate prediction of residual stress has significant impacts on structural fatigue and long-term performance. Eigenstrain reconstruction emerges as a competitive method for residual stress prediction, due to the appealing engineering adaptability and cost effectiveness. However, traditional eigenstrain methods, by polynomial forms, have difficulties in reconstructing residual stress with complex contour. In such cases, high-order polynomials have to be used to enhance reproduction capability within the domain. Unfortunately, polynomial interpolation with high degree over a set of measurements often encounters unexpected numerical oscillation, known as the Runge's phenomenon. This numerical limitation will misinterpret the residual stress with abrupt change or near the boundary of structures. To tackle these limitations, this paper investigates the residual stress prediction across dimensions. An eigenstrain reconstruction method based on radial basis function (RBF) is proposed in this work. By virtue of least squares method, full-field residual stress can be reproduced by solving an inverse eigenstrain problem through minimizing the residual errors between numerical predictions and experimental measurements. The radial basis function is used as the basis function space to reconstruct the full scale eigenstrain, and then the residual stress in three different dimensions. More importantly, a novel elliptical radial basis function has been proposed for predicting welding residual stress. Thanks to the unique scatter interpolation feature of radial basis function with limited experimental data, the distinctive advantage of the proposed method lies in the excellence of accurately predicting complex residual stresses in one dimension, the whole two dimensions or even three-dimensional components. The eigenstrain reconstruction method based on radial basis function enriches residual stress prediction with complex profiles in two-dimensional and three-dimensional space.

Optimal nonlinear polynomial control of a quarter-vehicle suspension system under harmonic and random road excitations

The present paper proposes an optimal polynomial control strategy for a quarter-vehicle suspension system based on dynamic programming. The optimal control objective is to decrease the responses of sprung mass acceleration, suspension deformation and road excitation, which are reflected in the performance index. The optimal control force is obtained through the principle of dynamic programming, and the form of the optimal control force mainly depends on the form of the cost function. The optimal nonlinear polynomial control (NPC) force is derived by selecting the cost function in polynomial form. The linear feedback matrix and nonlinear feedback matrix in the NPC force are determined by the Riccati equation and Lyapunov equation, respectively. The root mean square (RMS) responses of the sprung mass acceleration, suspension deformation and road load of driving vehicles under deterministic and random road surfaces are calculated. The numerical results showed that the proposed NPC strategy has a better control effect over the traditional linear quadratic regulator (LQR), especially on the peak reduction of sprung mass acceleration. Finally, the optimal control force is divided into active control force and passive control force. The energy input of the optimal control force can be reduced by only inputting the active control force.

Postbuckling Strengths of Composite Laminates with a Central Circular/Elliptical Cutout Under In-Plane Combined Loads

This paper deals with the effect of cutouts (i.e., circular and elliptical) aspect ratio on the buckling and postbuckling strengths, and failure characteristics of a simply-supported quasi-isotropic composite laminate subjected to in-plane shear and uni-axial compression combined with in-plane shear loads. The present study is carried out using self developed finite element program. The finite element formulation is based on the first order shear deformation theory and von Karmans assumptions to incorporate geometric nonlinearity. The resulting nonlinear algebraic equations are solved by Newton-Raphson method. The postbuckling strength of the laminate is defined in terms of the first-ply failure load of the lamina predicted by tensor polynomial form of the 3-D Tsai-Hill criterion. It is observed that for maximum buckling and first-ply failure strengths of the laminate with a cutout, it is preferred to have elliptical shape of the cutout as compared to the circular shape of the same area. It is also found that irrespective of load conditions and cutout shapes and sizes, shear load direction has significant effects on buckling and postbuckling stiffness of the laminate; although buckling stiffness is greater under negative shear, but the laminate shows stiffer postbuckling response under positive shear load.

Development of various machine learning and deep learning models to predict glycerol biorefining processes

Biorefining biological waste to produce eco-friendly fuels and by-products is essential in transitioning from non-renewable energies. However, the analysis of the processes in the laboratory necessitates a substantial investment of both time and money. The present study has developed machine learning (ML) models for evaluating biofuel products through glycerol biorefining for the first time. This study evaluates the performance of different ML algorithms, including recurrent neural network (RNN), random forest (RF), adaptive boosting (AdaBoost), Bayesian ridge (BR), and elastic net linear regression (ENLR). This research by machine learning algorithms can be created the formulas and the models for the hydrogen content (H2) by the hydrogen production and time, the hydrogen production by the H2 and time, OLR by PH and time, OLR by the hydrogen production and time, and the hydrogen production by OLR and time. Using the RNN for predicting the future of H2, hydrogen production, and OLR with the least error. The best R-Squared for the formulas is between 0.951 and 0.994 with the linear and the polynomial forms (by degrees 2, 3, 4). The best R-Squared for the models is between 0.998 and 0.999 with the linear form. MAEs for the formula and the model of H2, respectively, are 2.475347126 and 0.46588143, and MAEs for the formula and the model of the hydrogen production, respectively, are 23.44120285 and 7.03283978. MAEs of OLR by PH and OLR by the hydrogen production for formulas and models are 2.095157 and 000001 by PH and 3.148667 and 0.000001 by the hydrogen production. MAEs for the formula and the model of the hydrogen production by OLR, respectively, are 19.025255 and 2.718604.

Organic catalysts for hydrogen production from noodle wastewater: Machine learning and deep learning-based analysis

Hydrogen production from the electrolysis of wastewater is an environmentally friendly and highly efficient process. The performance of this process for instant noodle wastewater is strongly influenced by covering the PVC sheet with different arrangements of antioxidant-containing protein (ACAP) as an organic catalyst. However, analyzing this process through traditional models and experimental studies takes time, money, and effort. In the present research, several machine learning-based models, including the recurrent neural network (RNN), the least absolute shrinkage and selection operator (LASSO), the extreme gradient boosting (XGBoost), the linear regression (LR), and the light gradient-boosting machine (LightGBM) were developed to accurately predict hydrogen production performance from the electrolysis of noodle wastewater. Several materials have been studied in this research, such as Car, Car-Tur, Tur-Car, Tur, Car-Ver-Tur, and Car-Tof-Ver in the 12-V and 24-V states. For each material created, the LASSO regression and the linear regression formula include 12 formulas (six formulas for each state) for hydrogen production. The R-Squared values range between 0.989 and 0.997 for the six formulas by the polynomial form and by the XGBoost and the lightGBM making the six models for the hydrogen production, and the R-Squared values for all models are 0.999 by linear form for the hydrogen production in the 24-V state. For the hydrogen productions in the 12-V state, the values of the R-Squared range between 0.995 and 0.998 by the polynomial form. Using the lightGBM and the XGBoost, six models are made in linear form, and all of those models' R-Squared values are 0.999. Using the RNN algorithm can predict the future of each material's hydrogen production.

A novel adaptive approach for improvement in the estimation of hourly diffuse solar radiation: A case study of China

Liu & Jordan–type models are still widely used to estimate diffuse radiation in various regions because of their simplicity. This paper proposes an adaptive approach of combining the grouping of kt–k data points with Liu & Jordan–type models to estimate the hourly diffuse radiation more accurately. This study was conducted using hourly radiation data measured in three regions in China (Shanghai, Xi'an, and Lhasa). The main novelties reported herein are the introduction of α as a constraint indicator and the proposal of a criterion for grouping kt–k data points. Grouping models, which adopted the functional form of cubic polynomials, were established in three regions based on the grouped data. Compared with the all-data model results, the percentage of mean absolute bias error (MABE%) and relative root mean square error (RMSE%) for the grouping model decreased by approximately 4% in Shanghai and Xi'an and 6% in Lhasa. Furthermore, weather condition analysis showed that the grouping models had the best adaptability on overcast days, followed by cloudy days and sunny days. A similar analysis was performed for solar time, with the best adaptability occurring at sunrise and sunset and the worst occurring at 9 am. The results also suggest that similar working procedures could contribute to hourly diffuse radiation estimation in other fields.

A 〖(2n+9)〗^thDegree Polynomial Approximation Method for Solving Eight Order Linear Boundary Value Problems

Galerking method, Collocation method and least squired method have been used for solving boundary value problems (BVPs) and from practical point of view almost any success of above mention method depends on the selection of basis functions. Also, Adomain decomposition and variational iteration methods have been applied for the solution of BVPs and required respectively calculating the Adomian’s polynomials and Lagrange multipliers. In this paper a simple method of Degree polynomial approximation solution were n is number of successive derivatives of the governing equation is suggested as an alternative of the above mentioned methods. The method consists of first obtaining from the governing equation its n successive derivatives and boundary conditions a linear differential system of equations evaluated at the boundary points. Next, an approximate solution in the form of polynomial of degree with unknown coefficients is assumed. To determine the unknown coefficients, the assumed solution is incorporate into the linear differential system which turns to be a linear system of equation with unknowns which is salvable uniquely. The method is tested for n=5 to four examples. It clear that from tables 3.1 to 3.4 and figures 3.1 to 3.4 the numerical outcomes agree with the exact solution and also better than some existing results in the literatures. it should be also noted that the accuracy of the method can be increased by increased n. All Numerical computation were performed using maple 2016 software.

Quadratized Taylor series methods for ODE numerical integration

We focus on Taylor Series Methods (TSM) and Automatic Differentiation (AD) for the numerical solution of Ordinary Differential Equations (ODE) characterized by a vector field given by a finite composition of elementary and standard functions. We show that computational advantages are achieved if a kind of pre-processing said Exact Quadratization (EQ) is applied to the ODE before applying the TSM and the AD. In particular, when the ODE function is given by a formal polynomial (i.e. with real powers) of n variables and m monomials, the computational complexity required by our EQ based method for the calculation of the k-th order Taylor coefficient is O(k) whereas by using the existing AD methods it amounts to O(k2).

Symmetry and Asymmetry in Moment, Functional Equations, and Optimization Problems

The purpose of this work is to provide applications of real, complex, and functional analysis to moment, interpolation, functional equations, and optimization problems. Firstly, the existence of the unique solution for a two-dimensional full Markov moment problem is characterized on the upper half-plane. The issue of the unknown form of nonnegative polynomials on R×R+ in terms of sums of squares is solved using polynomial approximation by special nonnegative polynomials, which are expressible in terms of sums of squares. The main new element is the proof of Theorem 1, based only on measure theory and on a previous approximation-type result. Secondly, the previous construction of a polynomial solution is completed for an interpolation problem with a finite number of moment conditions, pointing out a method of determining the coefficients of the solution in terms of the given moments. Here, one uses methods of symmetric matrix theory. Thirdly, a functional equation having nontrivial solution (defined implicitly) and a consequence are discussed. Inequalities, the implicit function theorem, and elements of holomorphic functions theory are applied. Fourthly, the constrained optimization of the modulus of some elementary functions of one complex variable is studied. The primary aim of this work is to point out the importance of symmetry in the areas mentioned above.

Model and prediction relationship of sound velocity and porosity of seafloor sediments

The relationship between sound velocity and porosity in seafloor sediments is widely utilized for predicting or inverting sediment acoustical and physical parameters. This paper analyzes the theoretical relationship between sound velocity and porosity using the General Model of Sound Speed (GMSS). The GMSS model elucidates an empirical equation relationship in the form of a quadratic fitted polynomial based on actual measurement data from the South China Sea. The accuracy of the GMSS model and the regression empirical equation are found to be consistent, as demonstrated by the relative error, absolute error, and standard deviation of the sound velocity predictions. Our findings suggest that if the scattering of measured data is inherent to seafloor sediment, the empirical equation fails to adequately explain sediment variations and lacks broad applicability for accurate prediction and calculation. Additionally, we observe that a ± 5% relative error in porosity has minimal impact on the application of the GMSS model and empirical equations for predicting sound velocity errors. However, the empirical equation, relying solely on the porosity parameter, inadequately accounts for sound velocity scattering in seafloor sediments with the same porosity due to inherent limitations. In contrast, the GMSS model can incorporate additional physical parameters beyond porosity to better explain and predict sound velocity in seafloor sediments scattering in seafloor sediments with the same porosity, albeit with increased complexity. This paper provides an analytical method and theoretical foundation for inverting or predicting the relationship between sound velocity and porosity of seafloor sediments.

Nonlinear Identification through eXtended Outputs (NIXO) with numerical and experimental validation using geometrically nonlinear structures

This work presents a novel technique for nonlinear system identification that operates in the frequency domain and fits a model to measured spectra to estimate the parameters in a modal domain nonlinear equation of motion (EOM). Nonlinear terms are added to the linear EOM in the form of polynomials, and the proposed algorithm estimates the polynomial coefficients as well as the underlying linear Frequency Response Function (FRF). This method is an extension to a popular nonlinear system identification algorithm called NIFO, from Nonlinear Identification through Feedback of the Outputs. However, NIFO identifies the nonlinear parameters as complex numbers that may be different at each frequency line, even though the mechanical system is expected to be governed by an EOM in which the nonlinear parameters are real and constant with frequency. This might be problematic, because any variation in the identified nonlinear parameters will distort the linear FRFs estimated by NIFO, and those linear FRFs are important to tell the user whether all of the significant nonlinearity has been extracted from the system. The proposed algorithm, here dubbed Nonlinear Identification through eXtended Outputs (NIXO), estimates the nonlinear parameters as frequency-independent and real. Additionally, it is demonstrated that for the systems studied here that the algorithm works when random and swept-sine inputs are used to excite the tested structure, while NIFO only worked well when random inputs were used. The method is first evaluated numerically using benchmark case studies, starting with the SDOF equation and then reduced models of a clamped-clamped flat beam, and the results are compared to those obtained with NIFO. Then the algorithm is applied to swept-sine measurements from a 3D-printed flat beam and the results are validated by computing the primary nonlinear normal mode of the identified model and comparing it with measurements.

Formulas for Calculating the Deflections of a Triangular Truss of Spatial Cover

Statement of the problem. The scheme of triangular in terms of dome type cover is proposed. The construction is statically determinate. The formulas for the dependence of the deflection on the number of panels and sizes are derived by generalizing a series of individual solutions by induction. Materials and methods. The forces in the coating rods are performed by cutting nodes in symbolic form using operators of the Maple symbolic mathematics system. The unknown systems of equilibrium equations in projections on the coordinate axis include the reactions of vertical supports located on the sides of the truss. One of the corners of the truss also has a spherical support, one is cylindrical. The Maxwell—Mohr formula is used to calculate the deflection of the vertex. The analysis of sequences of coefficients in solutions for individual trusses with different numbers of panels yields expressions for common terms included in the desired calculation formula. Results. Formulas for the dependence of the deflections of the truss on the number of panels for a vertical load evenly distributed over the nodes of the truss and a horizontal wind load applied to one of the sides of the structure are obtained. The solutions have a simple polynomial form. The curves of the dependence of the horizontal displacement of the dome top on the number of panels reveal a minimum. Asymptotics of the solutions is identified. Conclusions. A scheme of a statically determinate symmetric spatial dome is developed and its mathematical model is constructed, allowing analytical solutions with an arbitrary number of panels. The identified dependencies can be used both to evaluate the accuracy of numerical solutions and to find optimal combinations of structural dimensions in terms of rigidity.

Comparison of Olympic-Style Weightlifting Performances of Elite Athletes: Scaling Models Account for Body Mass.

We developed a scale for comparison of performances by weightlifters of different body mass and compare this scaling formula to current systems. Data from Olympics and World and Continental Championships from 2017 to 2021 were obtained; results from athletes with doping violations were excluded, resulting in performances from 1900 athletes from 150 countries for use in analysis. Functional relationships between performance and body mass were explored by testing various transformations of body mass in the form of fractional polynomials that include a wide range of nonlinear relationships. These transformations were evaluated in quantile regression models to determine the best fit, examine sex differences, and distinguish fits for different performance levels (90th, 75th, and 50th percentiles). The resulting model used a transformation of body mass with powers -2 and 2 for males and females and was used to specify a scaling formula. The small percentage deviations between modeled and actual performances confirm the high accuracy of the model. In the subset of medalists, scaled performances were comparable across different body masses, whereas both Sinclair and Robi scalings, currently used in competitions, were more variable. The curves had similar shapes for the 90th and 75th percentile levels but were less steep for the 50th percentile. The scaling formula we derived to compare weightlifting performances across a range of body mass can easily be implemented in the competition software to determine the overall best lifters. This is an improvement over current methods that do not accurately account for differences in body mass and result in bias or yield large variations even with small differences in body mass despite identical performances.

New finite-type multi-indexed orthogonal polynomials obtained from state-adding Darboux transformations

Abstract The Hamiltonians of finite-type discrete quantum mechanics with real shifts are real symmetric matrices of order N + 1. We discuss the Darboux transformations with higher-degree (>N) polynomial solutions as seed solutions. They are state-adding and the resulting Hamiltonians after M steps are of order N + M + 1. Based on 12 orthogonal polynomials ((q-)Racah, (dual, q-)Hahn, Krawtchouk, and five types of q-Krawtchouk), new finite-type multi-indexed orthogonal polynomials are obtained, which satisfy second-order difference equations, and all the eigenvectors of the deformed Hamiltonian are described by them. We also present explicit forms of the Krein–Adler-type multi-indexed orthogonal polynomials and their difference equations, which are obtained from the state-deleting Darboux transformations with lower-degree (≤N) polynomial solutions as seed solutions.

A method for finding numerical solutions to Diophantine equations using Spiral Optimization Algorithm with Clustering (SOAC)

Diophantine equations are equations containing two or more unknowns, such that only the integer solutions are required. To find solutions of these equations numerically, we can be performed by solving an optimization problem using a metaheuristic method. In this paper, the Spiral Optimization Algorithm with Clustering (SOAC) method is proposed to find solutions to Diophantine equations in the form of polynomial, exponential, and also linear and nonlinear systems of equations. In the implementation of the method on solving some existing benchmark problems, the goal of simulation is to find all solutions only in a single run and in a short period of time. Appropriate values of required parameters are selected during the simulation. Results shows satisfactory in solving four problems in polynomial equations, four problems in exponential equations, and three problems in systems of linear and nonlinear equations. In most of cases, the results yield the same with the analytical or numerical solutions in the reference papers, and in some cases the results give more solutions.

Induced flow and heat transfer due to inner stretching and outer stationary coaxial cylinders

The current pioneering research explores the development of fluid flow and heat transfer between two cylinders. The flow is induced solely by the stretching of inner cylinder enclosed by the stationary outer cylinder. The physical phenomenon occurs in polymer extrusion in engineering applications. A mathematical formulation based on the mass conservation, Navier-Stokes and energy equations is designed which reveals after non-dimensionalization that the motion is governed by the cylinder curvature, Prandtl number and distance/gap parameters between the concentric cylinders. Numerical simulations portray that deformation of the inner cylinder sets up a motion drifting axially the fluid above, which is compensated by a radial fluid to conserve the mass flow. When the gap limits to infinity, the model perfectly matches with the Crane's solution for vanishing curvature and with the Wang's solution for all curvature. An analytical flow/heat solution in the specific case of narrowing cylinders is also derived in elementary polynomial form, which shows up as stretching triggered Coutte-kind flow. Yet, another analytical solution of the flow in terms of rational function valid with a specific value of curvature parameter is detected when the flow over the inner cylinder expands to infinity. In particular, the gap size has prominent effect on the formation of momentum and thermal fields. Results also prove that a critical gap distance exist at which the heat transfer rate is at its minimum. From a pure mathematical point of view, the present authors believe that some of the published numerical results should be rechecked through comparisons with the exact solutions as provided here.

Design of centrifugal radial fans using regression analysis methods

With the development of scientific and technological progress in agriculture, the use of operational and mathematical modelling for effective solution of problems and resource conservation in the field of agricultural engineering is relevant. Therefore, the purpose of the study was to determine the optimal parameters of the centrifugal radial fan of a pneumatic precision seed drill by constructing a new mathematical model of the process of its operation. This was achieved by applying mathematical modelling methods when planning multi-factor experiments. As a result, a complex of automated experiments has been defined, which leads to a significant increase in the productivity of scientific work. A statistical representation of the experiment is established, which allows moving to a multi-factor active experiment, in which it is possible to separate the influence of factors from the noise background and make a transition to statistical methods for analysing the results. This allowed predicting the optimal characteristics of the centrifugal radial fan of the precision seed drill. In the course of this study, a new regression equation was compiled in the form of a first-degree polynomial, which determines the influence of each of the factors on the magnitude and value of the response. The coefficients of the polynomial are determined, the significance of the coefficients is estimated, and the adequacy of the proposed model is checked. After obtaining the regression equation, it became possible to graphically construct the dependence of the response function on impact factors. A fractional factor experiment was also performed, which determined the values of the parameters of the object’s state for all possible combinations of levels of variation of the factors Xi. Based on the established functional relationship between the output parameter of the fan, a regression equation of the following form is obtained: Pv=Pv(n, ß1, ß2, z). This predicted the receipt of the total pressure Pv (Pa), when setting different values of independent quantities n, ß1, ß2 and z. The application of the obtained analytical dependencies significantly simplified the determination of optimal design parameters of pneumatic systems for the development and construction of modern technical seed drills